Group #28374471 — subgroups

Subgroups — #28374471

Order 13952.

370 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295631	C₂	1	2	yes	maximal, complemented
2	#20295327	C₂	1	4	yes	maximal, complemented
3	#20295279	C₂	1	2	yes	maximal, complemented
4	#20295262	C₂	1	2	yes	maximal, complemented
5	#20295222	C₂	1	4	yes	maximal, complemented
6	#424×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×D₈₇₂	C₂²	1	2	yes	—
8	C₂×D₈₇₂	C₂²	1	2	yes	—
9	#16812696	C₂²	1	2	yes	—
10	#16812691	C₂²	1	2	yes	—
11	#16812684	—	2	4	no	—
12	#16812673	C₂²	1	4	yes	—
13	#16812672	—	2	4	no	—
14	C₄×D₄₃₆	C₂²	1	4	yes	—
15	#16812633	C₂²	1	2	yes	—
16	#16812631	C₂²	1	2	yes	—
17	#16812621	C₂²	1	4	yes	—
18	#16812621	C₂²	1	4	yes	—
19	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
20	C₂²×C₈×C₁₀₉	C₂²	1	1	yes	characteristic
21	#117×C₁₀₉	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₀₉	C₂²	1	4	yes	—
23	C₂×D₄₃₆	—	2	4	no	—
24	C₂×D₄₃₆	C₂³	1	4	yes	—
25	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
26	C₂×C₄×D₁₀₉	—	2	4	no	—
27	D₈₇₂	—	4	2	no	—
28	D₈₇₂	—	4	2	no	—
29	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
30	#13182008	—	2	4	no	—
31	#13182006	—	2	4	no	—
32	#13182006	—	2	4	no	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	M₁₆×C₁₀₉	—	4	1	no	—
35	C₂×C₈×C₁₀₉	—	4	1	no	—
36	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
37	C₂×C₈×C₁₀₉	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₀₉	D₄	1	2	yes	—
39	C₂²⋊C₄×C₁₀₉	D₄	1	2	yes	—
40	C₄²×C₁₀₉	C₂³	1	2	yes	—
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂²×D₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	—	2	1	no	—
44	C₂×C₄×C₁₀₉	—	2	1	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
48	C₂²⋊D₁₀₉	—	8	4	no	—
49	C₂×Dic₁₀₉	—	2	4	no	—
50	D₄₃₆	—	2	4	no	—
51	D₄₃₆	—	2	4	no	—
52	D₄₃₆	—	8	4	no	—
53	C₄×D₁₀₉	—	8	4	no	—
54	C₈×C₁₀₉	—	2	2	no	—
55	C₈×C₁₀₉	—	2	2	no	—
56	C₂²×C₁₀₉	—	2	1	no	—
57	C₂²×C₁₀₉	—	2	2	no	—
58	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
59	C₂×D₁₀₉	—	4	4	no	—
60	C₂×D₁₀₉	—	4	4	no	—
61	C₂×D₁₀₉	—	4	4	no	—
62	C₄×C₁₀₉	—	2	1	no	—
63	C₄×C₁₀₉	—	2	4	no	—
64	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₁₀₉	—	4	4	no	—
67	C₂×C₁₀₉	—	4	1	no	—
68	C₂×C₁₀₉	#537	1	1	yes	characteristic
69	C₂×C₁₀₉	#576	1	2	yes	—
70	D₁₀₉	—	8	4	no	—
71	#3259	—	109	1	no	maximal, Sylow
72	C₁₀₉	#3259	1	1	yes	characteristic, complemented, Sylow
73	#540	—	109	2	no	—
74	#478	—	109	4	no	—
75	#469	—	109	2	no	—
76	#466	—	109	2	no	—
77	#459	—	109	4	no	—
78	#424	D₁₀₉	1	1	yes	characteristic, complemented
79	C₂×D₈	—	109	2	no	—
80	C₂×D₈	—	109	2	no	—
81	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₁₀₉	1	1	yes	characteristic
83	#127	—	109	2	no	—
84	#125	—	109	2	no	—
85	#123	—	218	4	no	—
86	#121	—	109	4	no	—
87	#121	—	218	4	no	—
88	C₄×D₄	—	109	4	no	—
89	#117	C₂×D₁₀₉	1	1	yes	characteristic
90	#107	—	109	2	no	—
91	#106	—	109	2	no	—
92	C₄⋊C₈	C₂×D₁₀₉	1	4	yes	—
93	#102	—	109	4	no	—
94	#102	—	109	4	no	—
95	C₂×D₄	—	109	4	no	—
96	C₂×D₄	—	218	4	no	—
97	C₂×D₄	—	218	4	no	—
98	C₂²×C₄	—	218	4	no	—
99	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
100	D₈	—	436	2	no	—
101	D₈	—	436	2	no	—
102	M₁₆	—	4	1	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
105	C₂×C₈	C₂²×D₁₀₉	1	2	yes	—
106	C₄⋊C₄	—	109	4	no	—
107	C₄⋊C₄	—	218	4	no	—
108	C₄⋊C₄	D₄₃₆	1	2	yes	—
109	C₂²⋊C₄	—	218	4	no	—
110	C₂²⋊C₄	—	218	4	no	—
111	C₂²⋊C₄	D₄₃₆	1	2	yes	—
112	C₄²	C₂²×D₁₀₉	1	2	yes	—
113	C₂³	—	218	4	no	—
114	C₂³	C₂×D₄₃₆	1	1	yes	characteristic
115	D₄	—	872	4	no	—
116	D₄	—	872	4	no	—
117	D₄	—	218	4	no	—
118	D₄	—	218	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	872	4	no	—
122	C₂×C₄	—	218	4	no	—
123	C₂×C₄	C₂×D₄₃₆	1	4	yes	—
124	C₂×C₄	C₂×D₄₃₆	1	1	yes	characteristic
125	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
126	C₈	—	2	2	no	—
127	C₈	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	436	4	no	—
131	C₂²	—	436	4	no	—
132	C₂²	—	436	4	no	—
133	C₂²	C₂²×D₄₃₆	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	2	1	no	—
136	C₄	—	436	4	no	—
137	C₄	#16812744	1	1	yes	characteristic
138	C₄	#16812747	1	1	yes	characteristic
139	C₂	—	4	1	no	—
140	C₂	—	872	4	no	—
141	C₂	#20295607	1	1	yes	characteristic
142	C₂	#20295831	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295631

C₂

yes

maximal, complemented

#20295327

C₂

yes

maximal, complemented

#20295279

C₂

yes

maximal, complemented

#20295262

C₂

yes

maximal, complemented

#20295222

C₂

yes

maximal, complemented

#424×C₁₀₉

C₂

yes

maximal, characteristic, complemented

C₂×D₈₇₂

C₂²

yes

—

C₂×D₈₇₂

C₂²

yes

—

#16812696

C₂²

yes

—

#16812691

C₂²

yes

—

#16812684

—

#16812673

C₂²

yes

—

#16812672

—

C₄×D₄₃₆

C₂²

yes

—

#16812633

C₂²

yes

—

#16812631

C₂²

yes

—

#16812621

C₂²

yes

—

#16812621

C₂²

yes

—

C₂×M₁₆×C₁₀₉

C₂²

yes

characteristic

C₂²×C₈×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₉

C₂²

yes

—

C₂×D₄₃₆

—

C₂×D₄₃₆

C₂³

yes

—

C₂×C₂²⋊D₁₀₉

—

C₂×C₄×D₁₀₉

—

D₈₇₂

—

D₈₇₂

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

#13182006

—

#13182006

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

M₁₆×C₁₀₉

—

C₂×C₈×C₁₀₉

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₂×C₈×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

D₄

yes

—

C₂²⋊C₄×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂²×D₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂²⋊D₁₀₉

—

C₂×Dic₁₀₉

—

D₄₃₆

—

D₄₃₆

—

D₄₃₆

—

C₄×D₁₀₉

—

C₈×C₁₀₉

—

C₈×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₉

2₋¹⁺⁴

yes

characteristic

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

#537

yes

characteristic

C₂×C₁₀₉

#576

yes

—

D₁₀₉

—

#3259

—

109

maximal, Sylow

C₁₀₉

#3259

yes

characteristic, complemented, Sylow

#540

—

109

—

#478

—

109

—

#469

—

109

—

#466

—

109

—

#459

—

109

—

#424

D₁₀₉

yes

characteristic, complemented

C₂×D₈

—

109

—

C₂×D₈

—

109

—

C₂×M₁₆

C₂×D₁₀₉

yes

characteristic

C₂²×C₈

C₂×D₁₀₉

yes

characteristic

#127

—

109

—

#125

—

109

—

#123

—

218

—

#121

—

109

—

#121

—

218

—

C₄×D₄

—

109

—

#117

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

C₄⋊C₈

C₂×D₁₀₉

yes

—

#102

—

109

—

#102

—

109

—

C₂×D₄

—

109

—

C₂×D₄

—

218

—

C₂×D₄

—

218

—

C₂²×C₄

—

218

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

100

D₈

—

436

—

101

D₈

—

436

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₁₀₉

yes

—

105

C₂×C₈

C₂²×D₁₀₉

yes

—

106

C₄⋊C₄

—

109

—

107

C₄⋊C₄

—

218

—

108

C₄⋊C₄

D₄₃₆

yes

—

109

C₂²⋊C₄

—

218

—

110

C₂²⋊C₄

—

218

—

111

C₂²⋊C₄

D₄₃₆

yes

—

112

C₄²

C₂²×D₁₀₉

yes

—

113

C₂³

—

218

—

114

C₂³

C₂×D₄₃₆

yes

characteristic

115

D₄

—

872

—

116

D₄

—

872

—

117

D₄

—

218

—

118

D₄

—

218

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

872

—

122

C₂×C₄

—

218

—

123

C₂×C₄

C₂×D₄₃₆

yes

—

124

C₂×C₄

C₂×D₄₃₆

yes

characteristic

125

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

436

—

131

C₂²

—

436

—

132

C₂²

—

436

—

133

C₂²

C₂²×D₄₃₆

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

436

—

137

C₄

#16812744

yes

characteristic

138

C₄

#16812747

yes

characteristic

139

C₂

—

140

C₂

—

872

—

141

C₂

#20295607

yes

characteristic

142

C₂

#20295831

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₄₃₆
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#424×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#424	D₁₀₉
hypocenter	C₁₀₉	#3259
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

subgroup

G/H

center

C₂²

C₂²×D₄₃₆

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#424×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ #117 ◁ #424

N	H ≅ G/N	type
1. #20295631	C₂	semidirect
2. #20295327	C₂	semidirect
3. #20295279	C₂	semidirect
4. #20295262	C₂	semidirect
5. #20295222	C₂	semidirect
6. #424×C₁₀₉	C₂	semidirect
72. C₁₀₉	#3259	semidirect
78. #424	D₁₀₉	semidirect

H ≅ G/N

type

1. #20295631

C₂

semidirect

2. #20295327

C₂

semidirect

3. #20295279

C₂

semidirect

4. #20295262

C₂

semidirect

5. #20295222

C₂

semidirect

6. #424×C₁₀₉

C₂

semidirect

72. C₁₀₉

#3259

semidirect

78. #424

D₁₀₉

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295631	2
#20295327	2
#20295279	2
#20295262	2
#20295222	2
#424×C₁₀₉	2
#3259	109

subgroup

index

#424×C₁₀₉

109

Subgroups — #28374471

Conjugacy classes of subgroups

Complements

Maximal subgroups